# Learning polytopes with fixed facet directions

arXiv: https://arxiv.org/abs/2201.03419 · HTML: https://mlsanity.com/paper/2201.03419/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 42 references that are in the map.
- First posted on arXiv 10 Jan 2022.

## Builds on (most cited first, 2 of 2)

- [Fitting Tractable Convex Sets to Support Function Evaluations](https://mlsanity.com/paper/1903.04194/) (2019)
- [Parametric shortest-path algorithms via tropical geometry](https://mlsanity.com/paper/1904.01082/) (2019)

## Built on by (most cited first, 1 of 1)

- [How to Learn a Star: Binary Classification with Starshaped Polyhedral Sets](https://mlsanity.com/paper/2505.01346/) (2025)

## Abstract (arXiv)

We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.